Why does X-mirror and Topology mirror hide mesh when all vertices are selected and translated? How can I view my mesh again?
Your mesh contains a lot of bad data, you should find that this bad data is causing the issues you are having.

With your object in edit mode, change to vertex select mode and de-select all vertices. With the mouse over part of the mesh press L - this will select all vertices linked to what is under the cursor. I see 748 vertices selected by this method.



I find that doing a select inverse and deleting the unconnected vertices still leaves some bad data. The following script deletes vertices that have one or more location values identified as NaN (Not a Number) which would be the reason you are having issues. After running this script I see 555 remaining vertices and find I can work with the remaining mesh and don't see any visual change.

import bpy, bmesh

obj = bpy.context.object
bm = bmesh.new()
bm.from_mesh(obj.data)
bad_verts = 0

for v in bm.verts:
    if isnan(v.co[0]) or isnan(v.co[1]) or isnan(v.co[2]):
        bad_verts += 1
        bm.verts.remove(v)

print('Removed {} bad vertices'.format(bad_verts))

bm.to_mesh(obj.data)
bm.free()